Stability and Regularity for Double Wall Carbon Nanotubes Modeled as Timoshenko Beams with Thermoelastic Effects and Intermediate Damping

This research studies two systems composed by the Timoshenko beam model for double wall carbon nanotubes, coupled with the heat equation governed by Fourier's law. For the first system, the coupling is given by the speed the rotation of the vertical filament in the beam $\beta\psi_t$ from the first beam of Tymoshenko and the Laplacian of temperature $\delta\theta_{xx}$, where we also consider the damping terms fractionals $\gamma_1(-\partial_{xx})^{\tau_1}\phi_t$, $\gamma_2(-\partial_{xx})^{\tau_2} y_t$ and $\gamma_3(-\partial_{xx})^{\tau_3} z_t$, where $(\tau_1, \tau_2, \tau_3) \in [0,1]^3$. For this first system we proved that the semigroup $S_1(t)$ associated to system decays exponentially for all $(\tau_1 , \tau_2 , \tau_3 ) \in [0,1]^3$. The second system also has three fractional damping $\gamma_1(-\partial_{xx})^{\beta_1}\phi_t$, $\gamma_2(-\partial_{xx})^{\beta_2} y_t$ and $\gamma_3(-\partial_{xx})^{\beta_3} z_t$, with $(\beta_1, \beta_2, \beta_3) \in [0,1]^3$. Furthermore, the couplings between the heat equation and the Timoshenko beams of the double wall carbon nanotubes for the second system is given by the Laplacian of the rotation speed of the vertical filament in the beam $\beta\psi_{xxt}$ of the first beam of Timoshenko and the Lapacian of the temperature $\delta\theta_{xx}$. For the second system, we prove the exponential decay of $S_2(t)$ for $(\beta_1, \beta_2, \beta_3) \in [0,1]^3$ and also show that $S_2(t)$ admits Gevrey classes $s>(\phi+1)/(2\phi)$ for $\phi=\min\{\beta_1,\beta_2,\beta_3\}, \forall (\beta_1,\beta_2,\beta_3)\in (0,1)^3$, and proving that $S_2(t)$ is analytic when the parameters $(\beta_1, \beta_2, \beta_3) \in [1/2,1]^3$. One of the motivations for this research was the work; Ramos et al. \cite{Ramos2023CNTs}, whose partial results are part of our results obtained for the first system for $(\tau_1, \tau_2, \tau_3) = (0, 0, 0)$.